Richard Nakka’s Experimental Rocketry Web Site

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Introduction to Rocket Design

Appendix K

Parachute Sizing and Loads

Introduction

My webpage Parachute Design and Construction provides a good introduction to the aerodynamics of parachutes, the basics of parachute design and testing, as well as means of constructing one’s own parachute. This appendix is intended to serve more as a design aid, providing details on sizing a parachute based on desired performance and expected loading conditions.

A bit of personal history. My first parachute equipped EX rocket was my “B” rocket that I first launched in April of 1972. Purchasing a suitable parachute back then was not an option. Consequently, I designed and made my own parachute using nylon fabric purchased at the local KMart. I fabricated my parachute with individual gores, alternating red and white fabric (perhaps not surprisingly, my inspiration was the Apollo parachute). The gores were sewn together using my mother’s sewing machine (with her guidance and approval, of course). Later on, I fortuitously came across some military surplus parachutes at the local Army Surplus store which were perfect for my rockets. Figure 1 illustrates a 33² circular chute (was white, I dyed it red) and a cross parachute (9²W ´ 28²L), both ripstop nylon. When that surplus supply dried up, and I needed a parachute for my Frostfire rocket, I decided to make a simple cross parachute. In more recent times, now that commercially made parachutes for rocketry have become available (and money not being as tight as it was), I purchase my parachutes. Fruity Chutes, for example, has a large selection of parachutes of all sizes and of different types, and the quality of their chutes is impeccable.

Figure 1: Two of my parachutes purchased at a military surplus store

Overview

Parachutes are, without a doubt, the ideal Recovery Descent system for rockets. A parachute can return a rocket to the ground more gently than any other recovery method. Even now in the 21^st century, parachutes remain the means of choice for recovery of commercial spacecraft such as Orion, Dragon and Starliner. The reasons are simple. Parachutes can be tailored to safely recover payloads of any size, from eggs to expended Space Shuttle Solid Rocket Boosters, weighing in at 175,000 pounds each. A parachute is lightweight, can be packed into a remarkably tiny volume, is generally inexpensive and durable. The desired rate of descent, or softness of landing, is simply a matter of choosing the appropriate size. The bigger the chute, the softer the landing. The principle of operation is simple making a parachute highly reliable. Figure2 illustrates the main parts of a typical EX parachute.

Figure 2: Parts of a typical parachute

When inflated, a parachute has a huge drag-to-mass ratio. Although a parachute is always used to descend some kind of payload, it is interesting to examine the Ballistic Coefficient of a parachute alone. The Ballistic Coefficient (see my webpage Aerodynamics and its Role in Experimental Rocket Design ) is given by:

where

m = mass

C_D = drag coefficient

A = cross-sectional area (normal to velocity vector)

As an example, the 42² (1067mm) ellipsoidal parachute I use for recovery of my Xi rocket has a mass of 120 grams (including shroud lines and bridle). The nominal cross-sectional, or projected, area is 894,000 mm². The C_D is 1.5 based on projected area.

As such, for this parachute:

A comparison of BC for different objects is given in this table. A ping-pong ball is 55´ as slippery as this parachute and the Xi rocket is 11,500´ as slippery ! (the more slippery an object, the less drag resistance it offers). As such, it is clear to see that attaching a parachute to a payload works well to slow its descent through the atmosphere.

Types of Parachutes

Parachutes, or rather the parachute canopy, come in a vast assortment of shapes and configurations. More common shapes are flat circular, conical, bi-conical, hemispherical, ellipsoidal, cross, annular, toroidal, ringslot, ringsail and ribbon-type. Ribbon-type or slotted chutes have particularly low opening forces. Parachutes can be employed singly or clustered. This webpage will focus on the three most common parachute types used for EX rocketry:

- Parasheet or flat circular

- Elliptical or ellipsoidal

- Cruciform or cross

I have made and used all three types over the years. The parasheet is simplest to make, it is basically a single piece of fabric cut to a circular shape. Shroud lines are sewn onto the hemmed circumference. This type of parachute is suited to model rockets and relatively lightweight LoPer class rockets. The Ellipsoidal and Cross type parachute are better performing (have a higher drag) and can be made to be robust enough to handle any weight rocket and to withstand greater opening velocity than a simple parasheet. The three canopy types are illustrated in Figure 3. The cross parachute is a good choice for deployment at high speed as it has a low opening shock factor. The cross parachute also performs well at supersonic velocities.

The shape of a parachute canopy when inflated is determined by a balance of internal pressure forces and tension in the shroud lines.

Figure 3: Canopy shape for various type parachutes

Notes: [1] Referenced to canopy surface area

[2] According to Res.K2, the drag coefficient greatly increases at low velocity

Drag Force of Parachutes

Parachute drag force is a key parameter, as it determines the rate of descent. Parachute drag will equal to the weight of the total assembly (rocket, tether and parachute). Parachute drag force, F_c, is given by:

C_D = parachute drag coefficient

S = reference area of canopy associated with C_D (ft² or m²)

q = ½ r v² = dynamic pressure due to descent velocity, v and r = local air density (slug/ft³ or kg/m³)

Units of drag, F_c, are either pounds-force (lbf) or Newtons (N).

Unlike streamlined bodies, Reynolds number does not change the drag coefficient of parachutes. This is because bodies with separated flow, such as parachute canopies, are little affected by Reynolds number.

As mentioned earlier, parachutes have a huge drag-to-mass ratio. Using the same example as earlier, the 42² (1067mm) ellipsoidal parachute that I use for recovery of my Xi rocket has a mass of 120 grams (including shroud lines and bridle). Drag coefficient is 1.5 (based on projected area). Descent rate for a 3 kg. Xi rocket is 20 ft/sec. or 6.1 m/sec. The parachute drag force is (using m.k.s system of units):

D = 1.5 π/4 (1.067)² [½ (1.225) (6.1)² ] = 30.6 N.

Giving a drag-to-mass ratio of 255 Newtons per kilogram, or in US units 26 pound-force per pound-mass.

Rate of Descent

A stable parachute in descent is in equilibrium between the total drag of the parachute and the weight of the payload. This is illustrated in Figure 4.

Figure 4: Forces acting on parachute in stable descent

The vertical descent rate (velocity) provided by a parachute in a stable descent is given by:

where Wt = total weight of body including parachute (kg-m/sec². or lb_m.), S = canopy reference area (m² or ft²), and r = local air density (kg/m³ or slug/ft³). The descent rate units are metres per second (m/s) or feet per second (fps).

Things to note about this equation.

1) Wt is weight, not mass. As such, if using SI system of units, mass must be multiplied by g, the acceleration of gravity (9.808 m/sec). i.e. W_t = m_t g

2) S is the reference area of the canopy upon which the drag coefficient, C_D, is based. Unfortunately, there exists inconsistency in definition of canopy area. Three possible reference areas are:

a. Nominal area. This is the cross-sectional area of a non-inflated canopy. For a circular parasheet, this is straightforward to calculate. However, for a shaped canopy (that does not lie flat) such as ellipsoidal, there is clearly some ambiguity.

b. Projected area. This is the cross-sectional projected area of the inflated parachute canopy. This is similar in concept to the reference area for solid bodies such as nose cones or rocket bodies. Manufacturers prefer this definition as it gives the highest numerical value of C_D. The problem with this reference area is that it is not constant. It changes with descent speed and shroud line length. As such, C_D referenced to parachute projected area should be specified in relation to inflated diameter (or other defining parameter), shroud line length and descent velocity.

c. Surface area of the canopy (S_o). This is the area of the actual fabric of the canopy. Engineers (such as T.W.Knacke) prefer this definition as the surface area of the canopy is fixed and as such in non-ambiguous. Vent holes and slots are included in this area, by definition. The drag coefficient based on surface (fabric) area indicates how effectively a parachute canopy produces drag with a given amount of fabric area. As such, drag coefficient referenced to canopy surface area (C_Do) provides an excellent measure of parachute efficiency (drag force per canopy mass).

3) Air density (r) decays with increasing altitude. Therefore descent velocity gradually lessens as the descending rocket gets closer to the ground, at a rate inversely proportional to the square root of r. This is one small, but not insigificant, way that nature is on the rocketeer’s side. The descent velocity (v_e) at any altitude is given by:

where v_eo is the descent velocity at sea-level, r_o is the air density at sea-level and r is the air density at the altitude of interest.

For a given parachute shape, drag coefficient can be increased by the use of longer shroud lines, greater number of shroud lines, and lower canopy porosity. Longer shroud lines increase the inflated diameter of the canopy, thereby increasing drag. This holds true for both circular cross-section and cross parachutes. My 42” ellipsoidal Fruity Chutes parachute has twelve shroud lines of 1.15´ chute diameter. Figure 5 shows the effect of shroud line length on the drag coefficient for a 40” cross parachute (taken from Res.K2). The increase in drag coefficient with shroud line length is quite significant. Note that the drag coefficient (CDo) for this example is referenced to canopy surface area (So = 2 L W – W ²).

Figure 5: Drag coefficient as function of shroud line length

It is worth noting that a higher drag coefficient does not necessarily mean a parachute with superior drag characteristics. It is the drag force that is the key measurement when comparing parachute shapes. For a given rate of descent, dynamic pressure is a fixed value. As such, it is the drag area (C_Do S_o ) that counts when comparing parachutes. Or more importantly, for weight critical applications such as a rocket, the drag force per unit mass of the parachute canopy. My 42” ellipsoidal Fruity Chutes parachute has a published C_D = 1.5. This is presumably based on canopy projected area.

Lower canopy porosity has the drawback that it can make a parachute less stable with a tendency to oscillate. Figure 6 illustrates the airflow around a porous and a non-porous canopy. As airflow cannot freely go through a non-porous canopy, the flow goes around and separates on the leading edge in alternating vortices, forming a Karman vortex trail. The separation results in alternate pressure areas on opposite sides of the canopy, generating an alternating force perpendicular to the airflow velocity vector, thereby generating oscillations.

Figure 6: Comparison between behaviour of porous vs non-porous canopy

In a similar manner to increasing porosity, adding a spill hole serves to reduce or eliminate such oscillation. The hole acts as a vent to release air pressure buildup without causing uneven air flow on one side of the canopy, and in doing so, stabilizes the descent by allowing trapped air to escape evenly, preventing the parachute from rocking or oscillating wildly as it falls, ensuring a more controlled and straight descent. A spill hole allows for use of a lower-porosity fabric. For example, my 42” ellipsoidal Fruity Chutes parachute has a spill hole that is 20% of the chute diameter. This allows for use of very low-porosity (3 ft³/ft²/min.) 1.2 oz. fabric (PIA-C 44378 type IV). For cross parachutes, such low canopy porosity results in rotational instability.

By rule of thumb, a spill hole area should be 1% of the canopy area. A spill hole is not necessary for a cross parachute.

Sizing the Parachute

The first step is to choose a suitable descent rate. For my rockets equipped with a drogue chute, descent from apogee to a lower altitude, a descent velocity in the range of 70-80 feet per second (21- 24 m/s), based on ground level air density, was considered appropriate. This has been considered as a good compromise between getting the rocket down quickly (to minimize downrange drift due to wind) and minimizing damage to the rocket in case the main chute fails to fully deploy. For descent under the main parachute, a suitable descent rate for my rockets has been in the range of 20-30 feet per second (6 – 9 m/s). The hardness of the terrain that the rocket will touchdown on is a factor. Less hard terrain such as tall grass or snow allows for a higher touchdown speed.

Knowing the total weight (W_t) of our descending rocket, and having settled on a descent speed (v), the surface area of the canopy (S_o) can be determined:

In this equation air density (r) should be that at ground level, whether sizing a drogue or main parachute. We know that the rocket with drogue chute will descend much faster at high altitudes, as air density drops with altitude. However, we wish to size the drogue chute based on descent velocity when either the main deploys (near the ground) or failing that, when the descending rocket reaches the ground.

It is important to note that the drag coefficient (C_Do) in equation 3 must be related to the canopy surface area as the reference area. If the drag coefficient is referenced to the canopy projected (or inflated) area, the equation is similar:

In this case, the canopy area, S, is taken as the projected area, as shown in Figure 7.

Figure 7: Canopy surface area versus projected area

Once the canopy surface area (S_o) has been determined, the canopy projected diameter (D_p) can be taken as the following:

Flat Circular diameter

Ellipsoidal base diameter

for b/a = 0.707, where a and b are the major and minor semiaxes, respectively

Cross panel length

where l = W/L

For a circular or ellipsoidal chute whereby the projected area has been calculated using equation 4:

Note that in all cases units must be consistent. For example, area in square feet gives diameter (or length) in feet.

Time of Descent

It is useful to know how long it would take for a rocket to descend from one altitude level (z₁) to lower altitude level (z₂). The time duration of descent can, for example, be used to estimate the downrange drift of a rocket, based on wind speed. Descent time, t_d, is given by:

assuming that the rocket descends at local terminal velocity (v_d). Using the drag force equation (Eqn.1), terminal velocity is given by

Atmospheric air density varies significantly with altitude. Using equation 9 and equation 10 together with polynomial curve fitting of the air density data results in the following expression for descent time between any two altitude points in the range of zero to 100,000 feet (30 km) above sea level (ASL):

with constant C₁ given by:

The coefficients A, B, and C are as follows:

US units of measure (slugs, feet): SI units of measure (m.k.s)

A = 1.057 e-12 2.582e-10

B = -3.756 e-7 -2.798e-5

C = 0.0490 1.112

(based on density lapse rate per International Standard Atmosphere)

Descent time calculated by this method is accurate to within approximately two percent based on the fidelity of the second-order curve fit. Details of the method are described here.

Canopy Opening

Parachute canopy inflation is a dynamic and chaotic (non-linear) process. The canopy expansion is resisted by tension in the shroud lines until fully inflated. The canopy inflation process is illustrated in Figure 8. The initial steps (a,b &c) occur relatively slowly, whereas the final steps (e,f &g) occur in rapid succession.

Figure 8: Parachute canopy inflation process

a) Opening of canopy mouth

b) Air mass moves into canopy

c) Ball of air reaches crown of canopy (less pronounced if spill hole present)

d) Influx of air expands the canopy

e) Expansion of canopy resisted by shroud line tension

f) Canopy becomes fully inflated

g) Canopy over-inflates; crown depressed by moment of surrounding air

I have had the good fortune of witnessing my fair share of parachute deployments and, as such, have observed this distictive sequence. In particular, how a parachute canopy opens slowly at first, then rapidly inflates, overinflates, then settles to its equilibrium shape. One additional behaviour has been observed. Following (g), it has been noticed that the canopy repeatedly overinflates then underinflates, doing this three or four times. This is strikingly visible in the video taken of the Xi-22 parachute (36²) deployment.

Xi-22 Parachute Deployment videoclip ( ¼ actual speed) 15 Mbytes

This cycle of overinflation/underinflation is certainly due to varying tension in the shroud lines. The inertia term in equation 12, acting in collaboration with the elastic nature of the parachute assembly, behaves as a spring-mass system.

The force (F) acting (at the bridle) of a vertically descending parachute plus rocket, as a function of time, is given by:

where:

C_D S = instantaneous drag area of the canopy (ft² or m²), which ranges from 0% to 100%

r = air density (slug/ft³ or kg/m³)

m_r = rocket mass (slug or kg)

m_p = parachute mass (slug or kg)

m_a = apparent mass (slug or kg)

The apparent mass is that of the air captured by the canopy. For most EX rockets, this term is much less than the mass of the rocket, and may be omitted. The middle term in equation 12 is the inertial force term due to the system mass and change in velocity of the rocket as it decelerates from free-fall (or drogue) descent velocity to parachute descent velocity .

The drag area (C_D S), as a function of time, can be taken as (see Res.K7):

where b = 6 for a regular (non-slotted) canopy, and t_f is the time to fully open. This relationship is plotted in Figure 9, showing percent of drag area versus time. As can be seen, the canopy opens at a slow rate initially, then suddenly blossoms to fully inflated state. How rapidly this occurs is, of course, dependent upon the size of the canopy, as well as other factors. When the Xi parachute (36” ellipsoidal) opens cleanly, this can occur is less than one second. It is this rapid opening that generates shock or opening force loading of the parachute.

Figure 9: Canopy drag area % vs time to fill ratio

It can be useful to know how long it takes for a canopy to fully inflate. When a rocket is descending, for example at free-fall velocity such as 85 feet per second (26 m/s.), each second the rocket gets 85 feet closer to the ground. The fill time needs to be considered when choosing a deployment altitude. From Res.K1, the canopy inflation time for a flat circular parachute may be expressed as:

where n = 4.0 for standard porosity and n = 2.5 for low-porosity canopies.

D_o is the nominal diameter given by:

For example, for the 36² ellipsoidal parachute for Xi-22 rocket, the canopy area S_o = 11.5 ft² , giving D_o = 3.82 ft. The free-fall velocity for this flight was 80.3 ft/sec., giving a canopy fill time of:

t_f = 4.0 (11.5)/(80.3)^0.85 = 1.1 second.

This assumes that the fill time for an ellipsoidal parachute is the same as for a flat circular chute. The value of 1.1 second is close to the fill time measured from the video, which is slightly less than one second. As such, the rocket descends approximately 80 feet from the time the chute is deployed to the time it fully inflates.

The deployment of a parachute generates two distinct and separate forces to the suspended load. These are known as the snatch force (Fs) and the opening force (Fx). Snatch force is the maximum shroud line tension during deployment prior to canopy inflation. When the parachute is deployed, the parachute and attached rocket must come to a common velocity. The inertial force of the rocket generates tension in the bridle when the shroud lines become taut. The canopy then inflates, followed by a brief overinflation, resulting in a peak momentary force that is transmitted to the rocket by the parachute. This is known as the opening force. The attachment force then decays to that of the equilibrium drag force (Fc). This sequence is illustrated in Figure 10.

Figure 10: Forces developed by a deployed and inflating parachute

Based on accelerometer data from many flights, it is apparent that the more critical of the two is the opening force (as per Fig.10). The equation for the opening force, taken from Res.K1, is:

C_x is the opening force coefficient assuming an infinite mass payload. This is the case where the mass of the payload is large compared to the drag force of the parachute. Values for C_x are:

Flat circular 1.7

Ellipsoidal 1.6

Cross 1.2

The force reduction factor, X₁, provides alleviation for the case where the payload mass is finite, or in the same order of magnitude (or so) as the parachute mass. This would be the case for EX rockets. The force reduction factor may be calculated using this graph with ballistic parameter, A, defined as:

For a flat circular, ellipsoidal, or cross parachute, the value of n = 2 should be assumed when interpreting the graph. The parameter is dimensionless, and as such, the following units are applicable:

Wt = total weight of body including parachute (kg-m/sec². or lb_m.), S = canopy reference area (m² or ft²), r = local air density (kg/m³ or slug/ft³), g = acceleration of gravity (9.81 m/sec² or 32.2 ft/sec²), v₁ = velocity at chute deploy (m/sec or ft/sec) and t_f = inflation time (sec.).

Note that the drag area term (C_D S) must be consistent. If C_D is referenced to the surface area of the canopy, then S = S_o. If C_D is referenced to the projected area of the canopy, then S is the projected area shown in Figure 7.

For my Xi rocket fitted with 42” ellipsoidal parachute, the force reduction factor works out to X₁ = 0.21. As such, the opening force comes to 31 lbf. (138 N.).

For design purposes, it is undoubtedly wise to assume the infinite mass condition and as such, conservatively assume X₁ = 1.

Resources

Res.K1 NWC TP 6575 Parachute Recovery Systems Design Manual, T.W.Knacke, Para Publishing

Res.K2 NOLTR 71-111 Effects of Canopy Geometry on the Drag Coefficient of a Cross Parachute in the Fully Open and Reefed Conditions W.P.Ludtke

Res.K3 MIL-C-7020 (Military Specification) Cloth, Parachute, Nylon-Rip Stop and Twill Weave

Res.K4 MIL-C-44378 (Military Specification) Cloth, Parachute, Nylon, Low Permeability (this spec is same as PIA-C-44378)

Res.K5 Parachute Canopy Fabric Mil Specs

Res.K6 FDL-TDR-64-155 Drag and Stability of Cross Type Parachutes, R.J.Niccum, E.L.Haak, & R.Gutenkauf

Res.K7AGARD-AG-295 The Aerodynamics of Parachutes D.J.Cockrell

Last updated January 23, 2025

Originally posted January 23, 2025

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